Global Estimation of the Cauchy Problem Solutions’ the Navier Stokes Equation

http://dx.doi.org/10.4236/jamp.2014.24003

Global Estimation of the Cauchy Problem

Solutions’ the Navier-Stokes Equation

A. A. Durmagambetov, L. S. Fazilova

Buketov Karaganda State University, Karaganda, Kazakhstan Email:[email protected]

Received 23 January 2014; revised 23 February 2014; accepted 28 February 2014

Copyright © 2014 by authors and Scientific Research Publishing Inc.

This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/

Abstract

The analytic properties of the scattering amplitude are discussed, and a representation of the po-tential is obtained using the scattering amplitude. A uniform time estimation of the Cauchy prob-lem solution for the Navier-Stokes equations is provided.

Keywords

Schrödinger’s Equation; Potential; Scattering Amplitude; Cauchy Problem; Navier-Stokes Equations; Fourier Transform

1. Introduction

This paper combines the results of studies on the inverse scattering problem with the Cauchy problem for the Navier-Stokes equations. First, we consider some ideas for the potential in the inverse scattering problem, and this is then used to estimate of solutions of the Cauchy problem for the Navier-Stokes equations. A similar ap-proach has been developed for one-dimensional nonlinear equations [1]-[4], but to date, there have been no re-sults for the inverse scattering problem for three-dimensional nonlinear equations. This is primarily due to diffi-culties in solving the three-dimensional inverse scattering problem.

This paper is organized as follows: first, we study the inverse scattering problem, resulting in a formula for the scattering potential. Furthermore, with the use of this potential, we obtain uniform time estimates in time of solutions of the Navier-Stokes equations, which suggest the global solvability of the Cauchy problem for the Navier-Stokes equations.

of a discrete spectrum, we obtain estimations for the maximum potential in the weaker norms, compared with the norms for Sobolev’ spaces.

Consider the operators H = −∆ +x q x

( )

, H0 = −∆x defined in the dense set

( )

2 3

2

W R in the space

( )

3

2

L R , and let q be a bounded fast-decreasing function. The operator H is called Schrödinger’s operator. We consider the three-dimensional inverse scattering problem for Schrödinger’s operator: the scattering potential must be reconstructed from the scattering amplitude. This problem has been studied by a number of researchers ([5]-[8] and references therein).

2. Results

Consider Schrödinger’s equation:

2

,

x q k k C

−∆ Ψ + Ψ = Ψ ∈ (1)

Let Ψ+

(

k, ,θ x

)

be a solution of (1) with the following asympotic behavior:

(

)

i ei

(

)

1

, , e , , 0 , ,

k x k x

k x A k x

x x

θ

θ θ θ

+

 

′

Ψ = + + _ _ → ∞

  (2)

where A k

(

, ,θ θ′

)

is the scattering amplitude and x, S2 x

θ′ = θ∈ for k∈C+=

{

Imk≥0

}

(

)

3

( ) (

)

i

1

, , , , e d .

4

k x

R

A k θ θ′ = − q x Ψ₊ kθ x − θ′ x

π

∫

(3)

Let us also define the solution Ψ₋

(

k, ,θ x

)

for k∈C−=

{

Imk≤0

}

as

(

k, ,θ x

)

(

k, θ,x

)

− +

Ψ = Ψ − −

As is well known [1]:

(

, ,

)

(

, ,

)

2

(

, ,

) (

, ,

)

d , .

4 S

k

kθ x kθ x A kθ θ kθ x θ k R

+ − ′ − ′ ′

Ψ − Ψ = − Ψ ∈

π

∫

(4)

This equation is the key to solving the inverse scattering problem, and was first used by Newton [6][7] and Somersalo etal. [8].

Equation (4) is equivalent to the following:

, S

+ −

Ψ = Ψ (5)

where S is a scattering operator with the kernel S k

( ) ( )

,ł S k ł, , _R3

( ) ( )

k x, ł x x, d ∗

+ −

=

_∫

Ψ Ψ .

The following theorem was stated in [1]:

Theorem 1 (The energy and momentum conservation laws)Let q∈R. Then, SS∗=I S S, ,∗ =I where I isaunitaryoperator.

Definition 1 The setofmeasurable functions R with thenorm, defined by R6

( ) ( )

2 d d

q x q y

q x y

x y

= < ∞

−

∫

R

isrecognizedasbeingofRollnikclass.

As shown in [8], Ψ±

( )

k x, is an orthonormal system of H eigenfunctions for the continuous spectrum. In addition to the continuous spectrum there are a finite number N of H negative eigenvalues, designated as

2

j

E

− with corresponding normalized eigenfunctions

(

_, 2

)

(

_1,

)

j x Ej j N

ψ − = , where

(

2

)

( )

3

2

,

j x Ej L R

ψ − ∈ .

We present Povzner’s results [9] below:

Theorem 2 (Completeness) For both an arbitrary

( )

3

2

f∈L R and for H eigenfunctions, Parseval’s identityisvalid.

(

) (

)

2

2

, , .

D D Ac Ac

L

(

)

1

, .

N

D j j j

j

P f fψ x E

=

=

∑

−

( ) (

)

2 2

0 , , d d ,

Ac _S

P f =

∫ ∫

∞ s f s Ψ+ sθ x θ s (6)

where f and fj are Fourier coefficients for the continuous and discrete cases.

Theorem 3 (Birmann-Schwinger estimation).Let q∈R. Then, thenumberofdiscreteeigenvaluescanbe estimatedas:

( )

( )

2 3 3

( ) ( )

2

1

d d . 4π R R

q x q y

N q x y

x y

≤

−

∫ ∫

(7)

This theorem was proved in [10]. Let us introduce the following notation:

(

)

(

)

2 , , d , for , , ,

S

NA=

∫

A k θ θ θ′ f = f k θ′ x

(

) (

)

2 , , , , d ,

S

Df =k

∫

A k θ θ′ f kθ′x θ′ (8)

(

)

i

0 , , e ,

z x

z x θ

φ θ =

(

)

(

(

)

i

)

, , , , e z x ,

zθ′x + z θ x θ

Φ = Ψ − ∆ (9)

where

(

) (

)

1

i i

N

j j

j

k E k E

=

∆ =

∏

+ − . We define the operators T±, T for f∈W21

( )

R as follows:

( )

0

1

lim d , 0,

2πiImz

f s

T f s Im z

s z ∞

+ _→

−∞

= >

−

∫

(10)

( )

0

1

lim d , 0,

2πiImz

f s

T f s Im z

s z ∞

− _→

−∞

= <

−

∫

(11)

(

)

1

. 2

Tf = T₊+T₋ f (12)

Consider the Riemann problem of finding a function Φ, that is analytic in the complex plane with a cut along the real axis. Values of Φ on the sides of the cut are denoted as Φ+, Φ−. The following presents the results of [11]:

Lemma 1

1 1 1 1 1

, , , , .

4 2 2 2 2

TT= I TT₊ = T₊ TT₋= − T T_{− +}= +T I T₋= −T I (13)

Theorem 4 Let q∈R, g= Φ − Φ

(

_{+ −}

)

. Then,

. T g

± ±

Φ = (14)

The proof of the above follows from the classic results for the Riemann problem.

Lemma 2 Let q∈R, , ,g₊=g

(

z θ x

)

, ,g₋=g

(

z −θ,x

)

. Then,

(

)

(

i

)

(

)

(

i

)

, , e z x , , , e z x .

z θ x T g θ zθ x T g − θ

+ + + − − −

Ψ ∆ = + Ψ ∆ = + (15)

The proof of the above follows from the definitions of g,Φ Ψ±, ±.

Lemma 3 Let q∈R,

(

, ,

)

, ,

(

,

)

.

(

, ,

)

(

)

.

A k θ θ′ ∆ =T+ A+∆ − ∆A− (16)

The proof of the above again follows from the definitions of the functions g,Φ Ψ±, ±.

Lemma 4 Let q∈R. Then,

(

)

.

NA+∆ =NT+ DA−∆ (17)

The proof of the above follows from the definitions of g,Φ Ψ±, ± and Theorem 1.

Definition 2 Denoteby  thesetoffunctions f k

(

, ,θ θ′

)

withthenorm sup _{, ,k}

(

)

.

TA

f = _{θ θ′} Tf + f < ∞

Definition 3 Denoteby _{(I DT− −)} thesetoffunctions g suchthat g=

(

I−DT−

)

f , forany f∈.

Lemma 5 Suppose A_TA< <α 1. Then, the operator

(

I−T D−

)

, defined on the set  has an inverse definedon ₍I T D−− ₎.

The proof of the above follows from the definitions of D T, − and the conditions of Lemma 5.

Lemma 6 Let q∈R, andassumethat

(

I−T D₋

)

−1 exists. Then,

g=T g+ −T g− (18)

(

)

1

0,

T g_{− −} = I−T D₋ − T D₋ φ (19)

(

)

1

0 0

1

. I T D − T Dφ φ

− − −

Ψ = − +

∆ (20)

The proof of the above follows from the definitions of g,Φ Ψ±, ± and Equation (4) Let us rewrite (20) using

(

K I

)

φ0.

± ±

Ψ = + (21)

Lemma 7 Let q∈R. Then,

1

, where .

F_±− =F_±′ F_±=K_±+I (22)

The proof is the same as that in [5].

Lemma 8 Let q∈R. Then,

0 0

lim .

z

q H _{− −}

→

= Ψ Ψ (23)

The lemma can be proved by substituting Ψ± into Equation (1).

Lemma 9 Let q∈R, andassumethat

(

I−T D₋

)

−1 exists. Then,

(

)

1

(

)

1

0 0 0 0

0

1 1

lim .

z

q= _→ _∆N I−T D− − T DH− φ   _∆N I−T D− − T D− φ +Nφ 

    (24)

The proof of the above follows from the definitions of N,Ψ± and Lemma 6.

Lemma 10 Let q∈R. Then D ≤2.

The proof of the above follows from the definition of D and the unitary nature of S.

Lemma 11 Let

( )

3

4

q∈RL R . Then,

( )

2 2

3 d ,

j _R j

E ≤

∫

q x ψ x (25)

( )

_{( )}

3

2

max j 2 _{j L R} .

x ψ x ≤ qψ (26)

The proof of the above follows from the definitions of E2j, ψj and (1).

Lemma 12 Let

( )

3

2

q∈RL R . Then,

( )

3

( )

2 ,

max_x P qD ≤2 q L R q Rmax_{x j} ψj x . (27)

The proof of the above follows from the definition of P fD .

Lemma 13 Let

( )

3

2

( )

3 2

max _Ac .

L R

x P q ≤C q (28)

To prove this result, one should calculate

(

)

3 3

2

d d

RqΨ =x R∆Ψ + Ψk x

∫

∫

(29) Using Lemma 7, the first approximation can be obtained in terms of q:

1

Ac

P q=_∆T Dq− +µ (30)

where µ represents terms of highest order of q. The lemma can be proved using obvious estimations for µ and Lemmas 8, 10.

3. Conclusions for the Three-Dimensional Inverse Scattering Problem

This study has shown once again the outstanding properties of the scattering operator, which, in combination with the analytical properties of the wave function, allow to obtain an almost-explicit formulas for the potential to be obtained from the scattering amplitude. Furthermore, this approach overcomes the problem of over- determination, resulting from the fact that the potential is a function of three variables, whereas the amplitude is a function of five variables. We have shown that it is sufficient to average the scattering amplitude to eliminate the two extra variables.

4. Cauchy Problem for the Navier-Stokes Equation

Numerous studies of the Navier-Stokes equations are devoted to the problem of the smoothness of its solutions. A good overview of these studies is given in [12]-[14]. The spatial differentiability of the solutions is an important factor, this controls their evolution.

Obviously, differentiable solutions do not provide an effective description of turbulence. On the other hand, the global solvability and differentiability of the solutions has not been proven, and therefore the problem of describing turbulence remains open.

It is interesting to study the properties of the Fourier transform of solutions of the Navier-Stokes equations. Of particular interest is how they can be used in the description of turbulence, and whether they are differentiable. The differentiability of such Fourier transforms appears to be related to the appearance or disappearance of resonance, as this implies the absence of large energy flows from small to large harmonics, which in turn precludes the appearance of turbulence.

Thus, obtaining uniform global estimations of the Fourier transform of solutions of the Navier-Stokes equations means that the principle modeling of complex flows and related calculations will be based on the Fourier transform method.

The authors are continuing to research these issues in relation to a numerical weather prediction model, and this paper is a theoretical justification for this approach.

Consider the Cauchy problem for the Navier-Stokes equations:

(

,

)

( )

, , 0,

t

q − ∆ +ν q q∇ = −∇ +q p f x t div q= (31)

( )

0 0

t

q ₌ =q x (32)

in the domain Q_T =R3×

( )

0,T , where:

0 0.

div q = (33)

The problem defined by (31), (32), (33) has at least one weak solution

(

q p,

)

in the so-called Leray-Hopf class [12].

The following results have been proved [12]:

Theorem 5 If

( )

( )

1 3

0 2 , 2 T ,

q ∈W R f∈L Q (34)

there is a single generalized solution of (31), (32), (33) in the domain 1

T

conditions:

( )

2

2

, , .

t T

q ∇ q ∇ ∈p L Q (35)

Note that T1 depends on q0 and f .

Lemma 14 Let 1

( )

3

0 2

q ∈W R , f∈L Q2

( )

T . Then,

( )

3

( )

3

( )

3 ( )

2 2 2 2

2

2 2

0

0 ₀

d .

sup

T

t

L R L R L R L Q

t T

q q τ q f

≤ ≤ + ∇

∫

≤ + (36)

Our goal is to provide global estimations for the Fourier transforms of derivatives of the Navier-Stokes equations’ solutions (31), (32), (33) without the that the smallness of the initial velocity and force are small. We obtain the following uniform time estimation. Using the notation that:

( )

( )

( )

₍

₎

_{( )}

( )

3 3

i , i ,

e k xd , e k l xd ,

R R

q k =

∫

q x x q k − =l

∫

q x − x (37)

( )

(

)

(

)

3

2 2

d ,

avg

R

q k =

∫

q k −l δ k −l l k (38)

Assertion 1 Thesolutionof (31), (32), (33) accordingtoTheorem 5 satisfies:

( )

(

₍

₎

)

2

0 0

e , d ,

t k t

q =q +

∫

−ν −τ _ q∇ q_+f τ (39)

where F = −∇ +p f .

This follows from the definition of the Fourier transform and the theory of linear differential equations.

Assertion 2 Thesolutionof (31), (32), (33) satisfies:

2 2

,

i j i

i j i

i j i

k k k

p q q i F

k k

=

∑

+

∑



  (40)

and the following estimations:

( )

3

( )

3

( )

3

2 2 2

3 1 2 2

3 ,

L R L R L R

p ≤ ∇q q (41)

2

2 2

1

3 .

f q

p f q

k _k k

∇ ≤  +  + ∇ + ∇  (42)

This expression for p is obtained using div and the Fourier transform. The estimations follow from this representation.

Lemma 15 Thesolutionof (31), (32), (33) inTheorem 5 satisfiesthefollowinginequalities:

3

2 2 2 2

3 0

d d d const,

t

R R

x q x+ x ∇q x τ≤

∫

∫ ∫

(43)

3 3

4 2 4 2

0

d d d const,

t

R R

x q x+ x ∇q x τ≤

∫

∫ ∫

(44)

or

( )

3

2 ₃

2 2

0

d d const,

t

L R R

q k q k τ

∇ +

_{∫ ∫}

∇ ≤ (45)

( )

3

2 3

2 2

2 2

0

d d const.

t

L R R

q k q k τ

∇  +

_{∫ ∫}

∇ ≤ (46)

Lemma 16 Thesolutionof (31), (32), (33) satisfiesthefollowinginequalities:

( )

3

( )

3

2 2

2 2

0

0 ₀

max max sup d ,

2

t

L R L R

k k _{t T}

T

q q q q τ

≤ ≤

≤ + + ∇

_∫

  (47)

( )

3

2 ₃

2 2 0

0 ₀

max max sup d d ,

2

t

L R

k k _{t T}

R

T

q q q k q k τ

≤ ≤

∇ ≤ ∇ + ∇ +

_{∫ ∫}

∇ (48)

( )

3

2 3

2 2

2 2 2 2

0

0 ₀

max max sup d d .

2

t

k k _{t T} L R

R

T

q q q k q k τ

≤ ≤

∇  ≤ ∇  + ∇  +

_{∫ ∫}

∇  (49)

These estimations follow from (9), Parseval’s identity, the Cauchy-Schwarz inequality, and Lemma 3.

Lemma 17 Thesolutionof (31), (32), (33) accordingtoTheorem 5 satisfies Ci≤const,

(

i=0, 2, 4 ,

)

where:

(

)

2

2 2 ₂

0 1 1 2 1 4 1

0 0 0

d , , , d , d .

t t t

C =

∫

F τ F = q ∇ +q F C = ∇

∫

F τ C = ∇

∫

F τ (50)

This follows from our a priori estimation (Lemma 1) and the assertion of Lemma 3.

Lemma 18 Thesolutionof (31), (32), (33) accordingtoTheorem 5 satisfiestothefollowinginequalities:

(

)

(

_k ,

)

q k e −e_λ t

 (51)

(

)

(

)

1 1 2 2 0 0 1 , 2 k k C

q k e e

k e e

λ λ ν   ≤ − _{+  } −  

 (52)

where

(

)

2 0 1 1

0

d , , .

t

C =

∫

F τ F = q∇ q+F (53)

Proof. From (39), we have the inequality:

(

)

(

)

(

(

)

)

2 2( )

₍

₍

₎

₎

0 1

0

, e k , d ,

t

k e e t

k k k

q k e −e_λ t ≤ q k e −e_λ +

∫

−ν −λ −τ F k e −e_λ τ τ (54)

where

(

)

1 , .

F = q∇ +q F (55)

Using the notation

( )

₍

₍

₎

₎

2 2

1 0

e k , d ,

t

k e e t

k

I =

∫

−ν −λ −τ F k e −e_λ τ τ (56)

and Hölder’s inequality in I , the following inequality can be obtained:

( ) 2 2 1 1 1 0 0

e k d d ,

t p p t q

q

k e e t

I ≤  −ν −λ −τ τ _{ } F τ_



∫

 

∫

 (57)

where p q, satisfy 1 1 1 p+ =q . Let p= =q 2. Then,

1 2 2 1 1 2 ₀ d 1 . 2 t k F I

k e e_λ

Using the estimation for I in (57), the assertion in the lemma can be proved. □

Lemma 19 Let q∈R, max

k q < ∞. Then,

( ) ( )

₍

₎

2 3 3

2

2 d d L max_k .

R R

q x q y

x y C q q

x y

≤ +

−

∫ ∫

 (59)

A proof of this lemma can be obtained using Plancherel’s theorem.

For 1 2 1 1 2 2 0 4π K CC ν ν = −

consider the transformation of the Navier-Stokes:

2

, , v, f .

t tA v f

A A A

ν ν

′= ′= ′= ′= (60)

Lemma 20 Let

(

)

1 ₂ 3 ₃ 0 4 1 A CC ν = +

, then 8. 7 K≤

Proof. Using the definitions for C и C0 we get

1 1 1 2 2 0 2 4πCC K

A A A

ν ν

−

 

   _{ }

=_{   } − _

   _{ } (61)

1 1 1 0 2 2 3 2

4π 8

. 7 CC A ν ν −    

= − <

 

 

(62)

□ We now obtain uniform time estimations for Rollnik’s norms of the solutions of (31), (32), (33). The follow- ing (and main) goal is to obtain the same estimations for max

x q—velocity components of the Cauchy problem

for the Navier-Stokes equations. We will use Lemmas 8 and 13.

Theorem 6 Let 2

( )

3

0 2 ,

q ∈W R ∇2q0∈L2

( )

R3 , f∈L Q2

( )

T ,

( )

( )

3

1 T 2 ,

f∈L Q L R

 _

( )

( )

2 3

1 T 2 .

f L Q L R

∇ ∈  Then, there exists a unique generalized solution of (31), (32), (33) satisfying the

followinginequality:

3 1

max max i const,

t x

i

q =

≤

∑

wherethevalueof const dependsonlyontheconditionsofthe theorem.

Proof. It suffices to obtain uniform estimates of the maximum velocity components qi, which obviously

follow from max i

x q , because uniform estimates allow us to extend the local existence and uniqueness theorem

over the interval in which they are valid. To estimate the velocity components, Lemma 12 can be used:

( )

3 2

2

0 0 d 1 ,

T

i i x L R

q =q  q t+A + 



∫



(

)

1 2

3 ₃

0 4 0 1 .

A = _ν CC + _

 

Using Lemmas (15)-(19) for

( )

3 2

2

0

0 d 1

T

i i x L R

q =q _ q t+A + _



∫



we can obtain A_{i TA}< <α 1, where Ai is the amplitude of potential qi and N q

( )

i <1. That is, discrete

solutions are not significant in proving the theorem, so its assertion follows the conditions of Theorem 6, which defines uniform time estimations for the maximum values of velocity components.

5. Conclusion

Uniform global estimations of the Fourier transform of solutions of the Navier-Stokes equations indicate that the principle modeling of complex flows and related calculations can be based on the Fourier transform method. In terms of the Fourier transform, under both smooth initial conditions and right-hand sides, no appear exacerbations appear in the speed and pressure modes. A loss of smoothness in terms of the Fourier transform can only be expected in the case of singular initial conditions, or of unlimited forces in L Q2

( )

T .

Acknowledgements

We are grateful to the Ministry of Education and Science of the Republic of Kazakhstan for a grant, and to the System Research “Factor” Company for combining our efforts in this project.

The work was performed as part of an international project, “Joint Kazakh-Indian studies of the influence of anthropogenic factors on atmospheric phenomena on the basis of numerical weather prediction models WRF (Weather Research and Forecasting)”, commissioned by the Ministry of Education and Science of the Republic of Kazakhstan.

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